On sets where lip f is finite

Buczolich, Zoltán and Hanson, Bruce and Rmoutil, Martin and Zurcher, Thomas (2019) On sets where lip f is finite. STUDIA MATHEMATICA, 249 (1). pp. 33-58. ISSN 0039-3223

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Abstract

Given a function f : R -> R, the so-called "little lip" function lip f is defined as follows:lip f(x) = lim inf(r SE arrow 0) sup(vertical bar x - y vertical bar <= r) vertical bar f(y) - f(x)vertical bar/r.We show that if f is continuous on R, then the set where lip f is infinite is a countable union of countable intersections of closed sets (that is, an F-sigma delta set). On the other hand, given a countable union E of closed sets, we construct a continuous function f such that lip f is infinite exactly on E. A further result is that, for a typical continuous function f on the real line, lip f vanishes almost everywhere.

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